Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $q \neq 0$. $a = \dfrac{q^2 + 8q + 16}{-2q^2 - 12q - 16} \div \dfrac{q - 3}{5q + 10} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{q^2 + 8q + 16}{-2q^2 - 12q - 16} \times \dfrac{5q + 10}{q - 3} $ First factor out any common factors. $a = \dfrac{q^2 + 8q + 16}{-2(q^2 + 6q + 8)} \times \dfrac{5(q + 2)}{q - 3} $ Then factor the quadratic expressions. $a = \dfrac {(q + 4)(q + 4)} {-2(q + 4)(q + 2)} \times \dfrac {5(q + 2)} {q - 3} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac { (q + 4)(q + 4) \times 5(q + 2)} { -2(q + 4)(q + 2) \times (q - 3)} $ $a = \dfrac {5(q + 4)(q + 4)(q + 2)} {-2(q + 4)(q + 2)(q - 3)} $ Notice that $(q + 4)$ and $(q + 2)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac {5\cancel{(q + 4)}(q + 4)(q + 2)} {-2\cancel{(q + 4)}(q + 2)(q - 3)} $ We are dividing by $q + 4$ , so $q + 4 \neq 0$ Therefore, $q \neq -4$ $a = \dfrac {5\cancel{(q + 4)}(q + 4)\cancel{(q + 2)}} {-2\cancel{(q + 4)}\cancel{(q + 2)}(q - 3)} $ We are dividing by $q + 2$ , so $q + 2 \neq 0$ Therefore, $q \neq -2$ $a = \dfrac {5(q + 4)} {-2(q - 3)} $ $ a = \dfrac{-5(q + 4)}{2(q - 3)}; q \neq -4; q \neq -2 $